The total work done in moving a particle along the curve is given by -----------

• A) $limit_{\delta->{\infty}}\sum F_{t} \delta_{s}=\int F_{t}ds$
• B) $limit_{\delta->{0}}\sum F_{t} \delta_{s}=\int F_{t}ds$
• C) $limit_{\delta->{0}}\sum W_{t} \delta_{s}=\int F_{t}ds$
• D) $limit_{\delta->{\infty}}\sum W_{t} \delta_{s}=\int F_{t}ds$

Integration along two separate paths joining the same two end points--------- give(s) identical results.

• A) Always
• B) Partially
• C) Necessarily
• D) Does not necessarily

$$\eqalign{ & {\text{Wallis sine formula when n is even}} \cr & \int\limits_0^{\frac{\pi }{2}} {Co{s^4}x} dx = \cr}$$

• A) $$\frac{3}{4} \cdot \frac{1}{2}$$
• B) $$\frac{4}{3} \cdot \frac{2}{1} \cdot \frac{\pi }{2}$$
• C) $$\frac{4}{5} \cdot \frac{2}{3}$$
• D) $$\frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi }{2}$$

$${\text{The}}\,\,{\text{div}}\,\,{\text{operator}}\,\,\nabla \,\,{\text{acts}}\,\,{\text{on}}\,\,{\text{a(an)}}\,\,{\text{_________}}\,\,{\text{and}}\,\,{\text{gives}}\,\,{\text{a}}\,\,{\text{scalar}}{\text{.}}$$

• A) $${\text{scalar}}$$
• B) $${\text{constant}}$$
• C) $${\text{unit}}\,\,{\text{vector}}$$
• D) $${\text{vector}}$$

Applications of Green’s Theorem are meant to be in -------

• A) two-dimensional
• B) One-dimensional
• C) three-dimensional
• D) four-dimensional

$${\text{If}} (Pdx + Qdy){\text{ is an exact differential}} {\text{then}}$$

• A) $$\oint {(Pdx + Qdy)} = 1$$
• B) $$\oint {(Pdx + Qdy)} \ne 0$$
• C) $$\oint {(Pdx + Qdy)} = 0$$
• D) $$\oint {(Pdx + Qdy)} = - 1$$

The path of integration of a line integral must be -------------

• A) continuous and single-valued
• B) straight and single-valued
• C) continuous and multi-valued
• D) straight and multi-valued

$$\begin{gathered} {\text{If}}\,\,{\text{a}}\,\,\_\_\_\_\_\_\_\_\_\,\,V(r)\,\,{\text{exists}}\,\,{\text{for}}\,\,{\text{all}}\,\,{\text{points}}\,\,{\text{on}}\,\,{\text{the}}\,{\text{curve,}}\,\,{\text{then}}\,\,\sum\limits_{p = 1}^n {V(r)\,d{r_p}} \,\,{\text{with}}\,\,dr \to 0\,\,{\text{defines}}\,\, \hfill \\ {\text{the}}\,\,{\text{line}}\,\,{\text{integral}}\,\,{\text{of}}\,\,V\,\,{\text{i}}{\text{.e}}{\text{.}}\,\,{\text{line}}\,\,{\text{integral}} = \int\limits_C {V(r)\,dr.} \hfill \\\ \end{gathered}$$

• A) $${\text{vector}}\,\,{\text{field}}$$
• B) $${\text{scalar}}\,\,{\text{field}}$$
• C) $${\text{vector}}\,\,{\text{quantity}}$$
• D) $${\text{vector}}\,\,{\text{space}}$$

$${\text{If}}\,\,Pdx + Qdy + Rdw\,\,{\text{is}}\,\,{\text{an}}\,\,{\text{exact}}\,\,{\text{differential}}\,\,{\text{equation}}\,\,{\text{then}}\,\,\_\_\_\_\_\_\_.$$

• A) $${\text{(d)}}\,\,\,\,\,{\text{All}}\,\,{\text{(a),}}\,\,{\text{(b)}}\,\,{\text{and}}\,\,{\text{(c)}}{\text{.}}$$
• B) $${\text{(a)}}\,\,\,\,\,\frac{{\partial P}}{{\partial y}} = \frac{{\partial Q}}{{\partial x}}$$
• C) $${\text{(c)}}\,\,\,\,\,\frac{{\partial R}}{{\partial y}} = \frac{{\partial Q}}{{\partial w}}$$
• D) $${\text{(b)}}\,\,\,\,\,\frac{{\partial P}}{{\partial w}} = \frac{{\partial R}}{{\partial x}}$$

Sign of line integral is reversed when -----------

• A) path of integration is divided into parts.
• B) path of integration is parallel to y-axis.
• C) direction of path of integration is reversed.
• D) path of integration is parallel to x-axis.

The path traversal in calculating the Green’s Theorem is --------

• A) outwards
• B) inwards
• C) clockwise
• D) anticlockwise

If a vector field [Math Processing Error]F(r)$F(r)$ exist for all points of the curve [Math Processing Error]C$C$, then we can form -----------------[Math Processing Error]F$F$ for each element of arc.

• A) vector Field
• B) scalar field
• C)
• D)

[Math Processing Error]IfI=∫ABPdx+Qdy and (Pdx+Qdy) is an exact differentialthen$${\text{If}} I = \int\limits_{AB} {Pdx + Qdy} {\text{ and }}(Pdx + Qdy){\text{ is an exact differential}} {\text{then}}$$

• A) [Math Processing Error]Ic1−Ic2=0$${I_{{c_1}}} - {I_{{c_2}}} = 0$$
• B) [Math Processing Error]Ic1×Ic2=0$${I_{{c_1}}} \times {I_{{c_2}}} = 0$$
• C) [Math Processing Error]Ic2+Ic2=0$${I_{{c_2}}} + {I_{{c_2}}} = 0$$
• D) [Math Processing Error]Ic1+Ic2=0$${I_{{c_1}}} + {I_{{c_2}}} = 0$$

\begin{gathered} {\text{If}}\,\,a > 0,\,\,{\text{then}}\,\,{\text{equations}}\,\,{\text{of}}\,\,{\text{the}}\,\,{\text{form:}} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{r^2} = {a^2}\cos 2\theta ,\,\,\,{r^2} = - {a^2}\cos 2\theta , \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{r^2} = {a^2}\sin 2\theta ,\,\,\,{r^2} = - {a^2}\sin 2\theta , \hfill \\ {\text{represent}}\,\,{\text{propeller - shaped}}\,\,{\text{curves}}\,\,{\text{called}}\,\,{\text{_________}}{\text{.}} \hfill \\\ \end{gathered}

• A) {\text{rose }}\,\,{\text{curve}}
• B) {\text{lemniscates}}
• C) {\text{spiral}}
• D) {\text{cardioids}}

{\text{In the integration in polar coordinates, }}dx\,dy{\text{ is replaced by - - - - - - - - - - - }}{\text{.}}

• A) dr\,d\theta
• B) d\theta
• C) r\,\,dr
• D) r\,dr\,d\theta

\begin{gathered} {\text{Let G be the rectangular box defined by the inequalities }}a \leqslant x \leqslant b,\,\,\,c \leqslant y \leqslant d,\,\,\,\,\,e \leqslant z \leqslant f. \hfill \ {\text{If }}f\,\,{\text{is continuous on G, then}}\,\,\int\limits_a^b {\int\limits_c^d {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dy\,\,dx = \,\,\, - - - - - - - - \hfill \\ \end{gathered}

• A) \int\limits_c^d {\int\limits_a^b {\int\limits_e^f {f(x,\,\,y,\,\,z)} } } \,dz\,\,dx\,\,dy\,
• B) \int\limits_a^b {\int\limits_e^f {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dz\,\,dx
• C) \int\limits_e^f {\int\limits_a^b {\int\limits_c^d {f(x,\,\,y,\,\,z)} } } \,dy\,\,dx\,\,dz
• D) {\text{All}}\,{\text{three}}\,{\text{options}}\,{\text{are}}\,{\text{true}}{\text{.}}

{\text{After}}\,{\text{reversing the order of limits of }}\,\int\limits_0^2 {\int\limits_x^2 {f(x,\,y)\,dy\,dx} } ,\,{\text{we}}\,{\text{get}}\, - - - - - -

• A) \int\limits_0^2 {\int\limits_x^2 {f(x,\,y)\,dx\,dy} }
• B) \int\limits_0^2 {\int\limits_0^y {f(x,\,y)\,dx\,dy} }
• C) \int\limits_x^2 {\int\limits_0^y {f(x,\,y)\,dy\,dx} }
• D) \int\limits_y^2 {\int\limits_x^2 {f(x,\,y)\,dx\,dy} }

{\text{If}} p(r,\theta ) {\text{is a point in polar coordinate system, then}} \theta {\text{is called}}

• A) {\text{Reflex angle of }}p
• B) {\text{Acute angle of }}p
• C) {\text{Reflex angle of }}p
• D) {\text{Polar angle of }}p

\int\limits_0^1 {\,\int\limits_0^{\ln \,y} {x\,} } dy\,dx\,\, = \,\, - - - - - - - - -

• A) \frac{{\ln \,y}} {2}
• B) \frac{{\ln \,x}} {2}
• C) \frac{{\left( {\ln \,x} \right)^2 }} {2}
• D) \ln \,y

{\text{In the integration of polar coordinates}} dxdy {\text{is replaced by}}

• A) d\theta
• B) drd\theta
• C) rdr
• D) rdrd\theta